Problem 106
Question
The function $$ f(x)=-0.00002 x^{3}+0.008 x^{2}-0.3 x+6.95$$ models the number of annual physician visits, \(f(x),\) by a person of age \(x .\) Graph the function in a \([0,100,5]\) by \([0,40,2]\) viewing rectangle. What does the shape of the graph indicate about the relationship between one's age and the number of annual physician visits? Use the TABLE or minimum function capability to find the coordinates of the minimum point on the graph of the function. What does this mean?
Step-by-Step Solution
Verified Answer
The minimum point of the function, which can be found using a graphing calculator, indicates the age at which a person has the least number of annual physician visits.
1Step 1: Graph the function
Begin by entering the function into a graphing calculator. Set the viewing window to [0,100,5] by [0,40,2] as advised. This will give an optimum view of the function within the given restrictions.
2Step 2: Interpret the graph
Observe the graph and take note of its shape. The graph gives the relationship between a person's age \(x\) and the number of their annual physician visits \(f(x)\). The graph's shape will indicate if the annual visits increase, decrease or remain stable as one gets older.
3Step 3: Find the minimum point
The minimum point of a graph can be found by seeing where the derivative equals zero and the second derivative is positive. Alternatively, if a graphing calculator is being used, it may have a 'minimum' function that will directly show the minimum point.
4Step 4: Interpret the minimum point
Once the minimum point is found, interpret what it means in the context of the problem. This will be the age at which the annual physician visits are least. The coordinates of the point give the age and the corresponding number of visits.
Key Concepts
College AlgebraPolynomial Function ModelingRelationship Between Age and Physician VisitsFinding Minimum Points in Functions
College Algebra
When tackling college algebra, one key skill students must develop is graphing polynomial functions. In the context of college algebra, graphing polynomials helps visualize relationships and interpret real-world scenarios through mathematical models.
For instance, a cubic polynomial in the form of \( f(x) = ax^3 + bx^2 + cx + d \) can represent a multitude of relationships, from physics to economics. The exercise provided shows a cubic polynomial used to model the number of an individual's physician visits based on age. Students learn to translate the coefficients and terms of the polynomial into meaningful data that are relevant to real-life situations. Understanding how to graph these functions is foundational in college algebra and serves as a precursor to more advanced studies, including calculus and beyond.
For instance, a cubic polynomial in the form of \( f(x) = ax^3 + bx^2 + cx + d \) can represent a multitude of relationships, from physics to economics. The exercise provided shows a cubic polynomial used to model the number of an individual's physician visits based on age. Students learn to translate the coefficients and terms of the polynomial into meaningful data that are relevant to real-life situations. Understanding how to graph these functions is foundational in college algebra and serves as a precursor to more advanced studies, including calculus and beyond.
Polynomial Function Modeling
Polynomial function modeling is a tool used to represent data and predict trends. It involves using polynomial equations to model phenomena, yielding insights that are otherwise hidden in raw data. These functions can be particularly useful in fields like biology, economics, and social sciences.
In our example, the polynomial \( f(x) = -0.00002x^3 + 0.008x^2 - 0.3x + 6.95 \) models the trend between age, \(x\), and annual physician visits, \(f(x)\). Such a model allows predicting how many physician visits a person might make, depending on their age. The coefficients in the polynomial give this model curve's shape, which can indicate an increase or decrease in visits at different age intervals.
In our example, the polynomial \( f(x) = -0.00002x^3 + 0.008x^2 - 0.3x + 6.95 \) models the trend between age, \(x\), and annual physician visits, \(f(x)\). Such a model allows predicting how many physician visits a person might make, depending on their age. The coefficients in the polynomial give this model curve's shape, which can indicate an increase or decrease in visits at different age intervals.
Relationship Between Age and Physician Visits
Discovering the relationship between age and the frequency of physician visits is crucial for healthcare planning and individual preventive care initiatives. The polynomial function provided in the exercise symbolizes this very relationship, giving a quantitative perspective on how these two factors are interconnected.
The graphing process of this function paints a picture, showing whether there's a tendency for the number of visits to increase as one gets older or if there's a certain age where these visits peak or decline. In interpreting the graph, we can find patterns such as whether children or the elderly require more frequent medical attention, which has critical implications for public health and policy-making.
The graphing process of this function paints a picture, showing whether there's a tendency for the number of visits to increase as one gets older or if there's a certain age where these visits peak or decline. In interpreting the graph, we can find patterns such as whether children or the elderly require more frequent medical attention, which has critical implications for public health and policy-making.
Finding Minimum Points in Functions
In the context of the polynomial function provided, finding the minimum number of visits can help identify an age where individuals are likely the healthiest or perhaps least likely to require medical attention. In mathematical terms, this is known as finding the minimum points in functions.
To find a minimum point, we often set the first derivative to zero and check if the second derivative is positive, ensuring it is indeed a minimum. With technology, like a graphing calculator, this process can be simplified using the 'minimum' function feature. Identifying the minimum point of our polynomial provides actionable data. For example, a minimum point at \((x, f(x))\) indicates that at the age of \(x\), a person is predicted to have the least number of physician visits \(f(x)\), according to the model.
To find a minimum point, we often set the first derivative to zero and check if the second derivative is positive, ensuring it is indeed a minimum. With technology, like a graphing calculator, this process can be simplified using the 'minimum' function feature. Identifying the minimum point of our polynomial provides actionable data. For example, a minimum point at \((x, f(x))\) indicates that at the age of \(x\), a person is predicted to have the least number of physician visits \(f(x)\), according to the model.
Other exercises in this chapter
Problem 105
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